Lattice models of statistical physics and a case study of box-crossing property

The thesis examines the theory of lattice models used in statistical physics from a mathematical perspective, and provides a case study of box-crossing property by proving Russo-Seymour-Welsh estimates in percolation model and the box-crossing property in the critical FK Ising (random cluster) model. The thesis is self-contained in its use of probability theory, but a general measure theory background is assumed.